Given $ m \angle CBD = 9x - 61$, $ m \angle ABC = 4x - 29$, and $ m \angle ABD = 131$, find $m\angle CBD$. $B$ $A$ $D$ $C$
From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {4x - 29} + {9x - 61} = {131}$ Combine like terms: $ 13x - 90 = 131$ Add $90$ to both sides: $ 13x = 221$ Divide both sides by $13$ to find $x$ $ x = 17$ Substitute $17$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 9({17}) - 61$ Simplify: $ {m\angle CBD = 153 - 61}$ So ${m\angle CBD = 92}$.